A module of a finite-dimensional simple Lie algebra is called a weight module if it is a direct sum of its weight subspaces. Let
\(M\)  be an irreducible highest-weight module for a finite-dimensional
simple Lie algebra \(\mathfrak {g}\) . Then \(M\)  is naturally a weight
module with finite-dimensional weight subspaces. Denote by \(d_{M}\) 
its Gelfand-Kirillov dimension. We fix a Cartan subalgebra
\(\mathfrak {h}\) , a root system \(\Delta \subset \mathfrak {h}^*\)  and a
set of positive roots \(\Delta _+ \subset \Delta \) . Let \(\rho \)  be half
the sum of all positive roots. Suppose that \(\beta \)  is the highest
root. It is well known that \(d_M=0\)  if and only if \(M\)  is
finite-dimensional, in which case irreducible modules are classified
by the highest-weight theory. From Vogan [1]} and Wang
[2]}, we know that the next smallest integer occurring is
\(d_M=(\rho ,\beta ^{\vee })\) . We call them the minimal
Gelfand-Kirillov dimension module. These small modules are of great
interest in representation theory. A general introduction can be
found in Vogan [1]}. Fernando [4]} showed that the
only simple Lie algebras which have simple torsion-free modules (or
cuspidal modules, i.e.,weight modules on which all root vectors of
the Lie algebra act bijectively) are those of types \(A_n\)  or \(C_n\) .
He also showed that these finitely generated torsion-free \(\mathcal {U}(\mathfrak {g})\) -modules have the minimal Gelfand-Kirillov
dimension. A similar result was independently obtained by Futorny
[5]}. In the last 20 years cuspidal modules have been
extensively studied by Benkart, Britten, Grantcharov, Hooper,
Khomenko, Lemire, Mathieu, Mazorchuk, Serganova and others (see e.g.
[6]},[7]}, [8]},[9]},
[10]}).
